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A129542
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Number of isolated primes < 10^n.
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2
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1, 10, 99, 820, 7145, 62161, 546620, 4880832, 43998523, 400227154, 3669302718, 33866741579, 314396207096, 2933381107473, 27490151938062, 258629969639330, 2441659478947916, 23122602510585989
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OFFSET
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1,2
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COMMENTS
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Isolated primes are primes that are not twin prime components. Define I(n) to be the number of isolated primes <= n. Given that Pi(n) -> infinity and I(n) -> infinity as n -> infinity, proving that pi(n) always grows by an ever so slight factor k>1 than I(n), then we will have infinity_Pi(n) - infinity_I(n) = infinity. So twin primes would be infinite in extent.
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LINKS
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C. Hilliard, Gcc code. It took 7.5 hrs to compute a(12). It will take the Gcc program 3.2 days to compute a(13). For a(16) it will take about 8 years.
Cino Hilliard, Sum of Isolated primes, message 38 in seqfun Yahoo group, providing code for gcc (needs formatting to become compilable), Jun 5, 2007. [Cached copy]
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FORMULA
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EXAMPLE
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The 10 isolated primes < 10^2 are 2,23,37,47,53,67,79,83,89,97 so 10 is the second entry in the table.
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PROG
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(PARI) countisoprimes(n) = \Count primes that are not twin prime components < 10^n { local(j, c, x); for(j=1, n, c=0; forprime(x=2, 10^j, if(!isprime(x-2)&&!isprime(x+2), c++) ); print1(c", ") ) }
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CROSSREFS
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KEYWORD
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hard,more,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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